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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Differentiation

JEE Advanced · Mathematics · Differentiation

If f(x)= _ n=1 ^ 50 (x-n)^ n(51-n) ; then f(51) f^ (51)

If f(x)=∏n=150(x−n)n(51−n)f(x)=\prod_{n=1}^{50}(x-n)^{n(51-n)}; then f(51)f′(51)\frac{f(51)}{f^{\prime}(51)}

Options

  1. A4040
  2. B1/1275
  3. C1/4040
  4. D1275

Correct answer

B. 1/1275

Step-by-step solution

f(x)=∏n=150(x−n)n(51−n)ln⁡(f(x))=∑n=150n(51−n)ln⁡(x−n)\begin{aligned} & f(x)=\prod_{n=1}^{50}(x-n)^{n(51-n)} \\ & \ln (f(x))=\sum_{n=1}^{50} n(51-n) \ln (x-n) \end{aligned} Differentiating both side w.r.t. xxf′(x)f(x)=∑n=150n(51−n)x−n⇒f′(51)f(51)=1275\begin{aligned} & \frac{f^{\prime}(x)}{f(x)}=\sum_{n=1}^{50} \frac{n(51-n)}{x-n} \\ & \Rightarrow \frac{f^{\prime}(51)}{f(51)}=1275 \end{aligned}$$
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